#### Solved Examples and Worksheet for Factoring Quadratic Expressions

Q1Factor:
42x2 - 71x + 30

A. (6x - 6)(7x - 6)
B. (6x + 7)(7x + 6)
C. (6x - 5)(7x - 6)
D. (x + 5)(42x - 30)

Step: 1
42x2 - 71x + 30
[Given trinomial.]
Step: 2
a = 42, b = - 71, c = 30
[Compare with ax² + bx + c.]
Step: 3
Find two numbers whose product is 42 and another two numbers whose product is 30.
Factors of 42          Factors of 30
1, 42 and 6, 7          - 5, - 6
Step: 4
Use FOIL method to check the middle term in the trial factors.
Trial factors           Middle term
(x - 5)(42x - 6)     - 6x - 210x = - 216x
(6x - 5)(7x - 6)    - 36x - 35x = - 71x
Step: 5
So, 42x2 - 71x + 30 = (6x - 5)(7x - 6).
Correct Answer is :   (6x - 5)(7x - 6)
Q2Factor the trinomial.
3y2 - 10y - 8

A. (y - 2)(3y + 4)
B. (y + 3)(2y + 2)
C. (y - 1)(3y - 8)
D. (y - 4)(3y + 2)

Step: 1
3y2 - 10y - 8
[Given trinomial.]
Step: 2
a = 3, b = - 10, c = - 8
[Compare with ax² + bx + c.]
Step: 3
Find two numbers whose product is 3 and another two numbers whose product is - 8.
Factors of 3       Factors of - 8
1, 3           - 1, 8; - 2, 4 and - 4, 2
Step: 4
Use FOIL method to check the middle term in the trial factors.
Trial factors           Middle term
(y - 1)(3y + 8)      8y - 3y = 5y
(y - 2)(3y + 4)     4y - 6y = - 2y
(y - 4)(3y + 2)     2y - 12y = - 10y
Step: 5
So, 3y2 - 10y - 8 = (y - 4)(3y + 2).
Correct Answer is :   (y - 4)(3y + 2)
Q3Factor the trinomial.
3x2 - 10xy - 25y2

A. (x + 5y)(3x - 5y)
B. (3x + 5y)(x - 5y)
C. (x - 5y)(x - 5y)
D. (x + y)(3x + 25y)

Step: 1
3x2 - 10xy - 25y2
[Given trinomial.]
Step: 2
a = 3, b = - 10, c = - 25
Step: 3
Find two numbers whose product is 3 and another two numbers whose product is - 25.
Factors of 3        Factors of - 25
1, 3            1, - 25 and 5, - 5
Step: 4
Use FOIL method to check the middle term in the trial factors.
Trial factors           Middle term
(x + y)(3x - 25y)      - 25xy + 3xy = - 22xy
(x + 5y)(3x - 5y)      - 5xy + 15xy = 10xy
(3x + 5y)(x - 5y)     - 15xy + 5xy = - 10xy
Step: 5
So, 3x2 - 10xy - 25y2 = (3x + 5y)(x - 5y).
Correct Answer is :   (3x + 5y)(x - 5y)
Q4Factor:
x2 + 3x  - 10

A. (x - 1)(x + 10)
B. (x - 2)(x + 5)
C. (x + 1)(x - 10)
D. (x + 3)(x - 10)

Step: 1
x2 + 3x - 10 is in the form of x2 + bx + c. So, b = 3 and c = - 10.
Step: 2
x2 + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.
Step: 3
 p and q p + q p × q 1, - 10 - 9 - 10 - 2, 5 3 - 10
[Select the values of p and q by trial and error.]
Step: 4
The required values of p and q are - 2, 5.
[p + q = 3, pq = - 10.]
Step: 5
Therefore, x2 + 3x - 10 = (x - 2)(x + 5)
[Substitute for p and q.]
Correct Answer is :   (x - 2)(x + 5)
Q5Factor:
x2 + 13x + 40

A. (x + 20)(x + 2)
B. (x + 13)(x + 40)
C. (x - 1)(x - 40)
D. (x + 8)(x + 5)

Step: 1
x2 + 13x + 40 is in the form of x2 + bx + c. So, b = 13 and c = 40.
Step: 2
x2 + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.
Step: 3
 p and q p + q p × q 1, 40 41 40 20, 2 22 40 8, 5 13 40
[Select the values of p and q by trial and error.]
Step: 4
The required values of p and q are 8, 5.
[p + q = 13, pq = 40.]
Step: 5
Therefore, x2 + 13x + 40 = (x + 8)(x + 5).
[Substitute for p and q.]
Correct Answer is :   (x + 8)(x + 5)
Q6Factor:
y2 - 3y - 70

A. (y + 17)(y - 70)
B. (y - 1)(y - 70)
C. (y - 3)(y - 70)
D. (y - 10)(y + 7)

Step: 1
y2 - 3y - 70 is in the form of x2 + bx + c. So, b = - 3 and c = - 70.
Step: 2
x2 + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.
Step: 3
 p and q p + q p × q 1, - 70 - 69 - 70 - 10, 7 - 3 - 70
[Select the values of p and q by trial and error.]
Step: 4
The required values of p and q are - 10, 7.
[p + q = - 3, pq = - 70.]
Step: 5
Therefore, y2 - 3y - 70 = (y - 10)(y + 7).
[Substitute for p and q.]
Correct Answer is :   (y - 10)(y + 7)
Q7A rectangular playground is about 6 ft longer than it's width. The area of the ground is 55 sq.ft. Find the length.

A. 5 ft
B. 16 ft
C. 12 ft
D. 11 ft

Step: 1
Let the width of the ground = x ft.
Step: 2
Length of the ground = (x + 6) ft.
Step: 3
Area of the ground = 55 sq.ft.
Step: 4
So, x(x + 6) = 55
[Area of a rectangle = length × width.]
Step: 5
x2 + 6x = 55
Step: 6
x2 + 6x - 55 = 0
Step: 7
The factors of a trinomial x2 + bx + c are in the form (x + p)(x + q), where b = p + q and c = pq.
Step: 8
Compare the left side of the equation with x2 + bx + c to get b and c values. So, b = 6 and c = - 55.
Step: 9
Find the numbers p and q whose product is - 55 and whose sum is 6.
Step: 10
 p and q p + q p × q - 11, 5 - 6 - 55 11, - 5 6 - 55
[Select the values of p and q by trial and error.]
Step: 11
The required values of p and q are 11, - 5.
[p + q = 6 , pq = - 55.]
Step: 12
x2 + 6x - 55 = (x + 11)(x - 5)
[Substitute for p and q.]
Step: 13
So, the equation x2 + 6x - 55 = 0 can be written as (x + 11)(x - 5) = 0
Step: 14
x + 11 = 0 or x - 5 = 0
Step: 15
x = - 11 or x = 5
Step: 16
The width of the ground = 5 ft.
[Negative values for dimension do not make sense.]
Step: 17
The length of the ground = 5 + 6 = 11 ft.
Correct Answer is :   11 ft
Q8Factor: x2 + 6x + 9
A. (x + 3)(x - 3)
B. (x + 3)(x + 3)
C. (x - 4)(x + 3)
D. (x - 3)(x - 3)

Step: 1
The factors of a trinomial x2 + bx + c are in the form (x + p)(x + q), where b = p + q and c = pq.
Step: 2
Compare the equation with x2 + bx + c to get b and c values. So, b = 6 and c = 9.
Step: 3
Find the numbers p and q whose product is 9 and sum is 6.
Step: 4
p and q    p + q
1, 9        10
3, 3          6
Step: 5
The required values of p and q are 3 and 3.
Step: 6
So, x2 + 6x + 9 = (x + 3)(x + 3).
Correct Answer is :   (x + 3)(x + 3)
Q9Factor:
x2 - 4x + 3

A. (x + 1)(x - 3)
B. (x + 1)(x + 3)
C. (x - 1)(x - 3)
D. (x - 1)(x + 3)

Step: 1
x2 - 4x + 3 is in the form of x2 + bx + c. So, b = - 4 and c = 3.
Step: 2
x2 + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.
Step: 3
 p and q p + q p × q - 1, - 3 - 4 3
[Select the values of p and q by trial and error.]
Step: 4
The required values of p and q are - 1 and - 3.
[p + q = - 4, pq = 3.]
Step: 5
So, x2 - 4x + 3 = (x - 1)(x - 3).
[Substitute for p and q.]
Correct Answer is :   (x - 1)(x - 3)
Q10Factor: x2 + 2x - 15

A. (x - 3)(x - 5)
B. (x + 3)(x + 5)
C. (x + 3)(x - 5)
D. (x - 3)(x + 5)

Step: 1
The factors of a trinomial x2 + bx + c are in the form (x + p)(x + q), where b = p + q and c = pq.
Step: 2
Compare the equation with x2 + bx + c to get b and c values. So, b = 2 and c = -15.
Step: 3
Since c is negative, find the numbers p and q with different signs, whose product is -15 and sum is 2.
Step: 4
p and q    p + q
-1, 15         14
-3, 5           2
Step: 5
The required values of p and q are -3 and 5.
Step: 6
So, x2 + 2x - 15 = (x - 3)(x + 5).
Correct Answer is :   (x - 3)(x + 5)
Q11Factor:
x2 - 4x - 21

A. (x - 7)(x - 3)
B. (x + 7)(x + 3)
C. (x + 7)(x - 3)
D. (x - 7)(x + 3)

Step: 1
x2 - 4x - 21 is in the form of x2 + bx + c. So, b = - 4 and c = - 21.
Step: 2
x2 + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.
Step: 3
 p and q p + q p × q - 3, 7 4 - 21 - 7, 3 - 4 - 21
[Select the values of p and q by trial and error.]
Step: 4
The required values of p and q are - 7 and 3.
[p + q = - 4, pq = - 21.]
Step: 5
So, x2 - 4x - 21 =(x - 7) (x + 3).
[Substitute for p and q.]
Correct Answer is :   (x - 7)(x + 3)
Q12Factor: x2 - 3x - 28
A. (x + 7)(x - 4)
B. (x - 7)(x - 4)
C. (x - 7)(x + 4)
D. (x + 7)(x + 4)

Step: 1
The factors of a trinomial x2 + bx + c are in the form (x + p)(x + q), where b = p + q and c = pq.
Step: 2
Compare the equation with x2 + bx + c to get b and c values. So, b = -3 and c = -28.
Step: 3
Since c is negative, find the numbers p and q with different signs, whose product is -28 and sum is -3.
Step: 4
p and q    p + q
- 4, 7           3
- 7, 4         - 3
Step: 5
The required values of p and q are -7 and 4.
Step: 6
So, x2 - 3x - 28 = (x - 7)(x + 4).
Correct Answer is :   (x - 7)(x + 4)
Q13Factor: x2 - 7x + 10
A. (x + 5)(x + 2)
B. (x - 5)(x - 2)
C. (x - 5)(x + 2)
D. (x + 5)(x - 2)

Step: 1
The factors of a trinomial x2 + bx + c are in the form (x + p)(x + q), where b = p + q and c = pq.
Step: 2
Compare the equation with x2 + bx + c to get b and c values. So, b = -7 and c = 10.
Step: 3
Since c is positive, find the numbers p and q with the same sign, whose product is 10 and sum is -7.
Step: 4
p and q    p + q
-5, -2           -7
Step: 5
The required values of p and q are -5 and -2.
Step: 6
So, x2 - 7x + 10 = (x - 5)(x - 2).
Correct Answer is :   (x - 5)(x - 2)
Q14Factor:
x2 + 5x + 6

A. (x - 1)(x - 6)
B. (x + 2) (x + 3)
C. (x + 5)(x + 6)
D. (x - 2)(x + 3)
E. (x + 1)(x + 6)

Step: 1
x2 + 5x + 6 is in the form of x2 + bx + c. So, b = 5, c = 6.
Step: 2
x2 + bx + c can be factored as (x + p) (x + q), where b = p + q and c = pq.
Step: 3
p and q      p + q      p × q
Step: 4
1, 6           7           6
Step: 5
2, 3              5            6
Step: 6
The required values of p and q are 2 and 3.
Step: 7
Therefore, x2 + 5x + 6 = (x + 2)(x + 3).
Correct Answer is :   (x + 2) (x + 3)
Q15Factor.
x2 - 11x + 24

A. (x + 8)(x - 5)
B. (x - 11)(x + 5)
C. (x - 3)(x - 8)
D. (x - 1)(x - 24)

Step: 1
x2 - 11x + 24 is in the form of x2 + bx + c. So, b = - 11 and c = 24.
Step: 2
x2 + bx + c can be factored as (x + p)(x + q), where b = p + q and c = pq.
Step: 3
 p and q p + q p × q - 1, - 24 - 25 24 - 3, - 8 - 11 24
[Select the values of p and q by trial and error.]
Step: 4
The required values of p and q are - 3, - 8.
[p + q = - 11, pq = 24.]
Step: 5
Therefore, x2 - 11x + 24 = (x - 3)(x - 8).
[Substitute for p and q.]
Correct Answer is :   (x - 3)(x - 8)